MinT - Best Known (196−68, 196, s)-Nets in Base 4

Best Known (196−68, 196, s)-Nets in Base 4

(196−68, 196, 160)-Net over F₄ — Constructive and digital

Digital (128, 196, 160)-net over F₄, using

1 times m-reduction [i] based on digital (128, 197, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 47, 30)-net over F₄, using
    - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
        F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (81, 150, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 75, 65)-net over F₁₆, using
      - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(196−68, 196, 208)-Net in Base 4 — Constructive

(128, 196, 208)-net in base 4, using

2 times m-reduction [i] based on (128, 198, 208)-net in base 4, using
- trace code for nets [i] based on (29, 99, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (29, 100, 104)-net in base 16, using
    - base change [i] based on digital (9, 80, 104)-net over F₃₂, using
      - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(196−68, 196, 468)-Net over F₄ — Digital

Digital (128, 196, 468)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(196−68, 196, 13309)-Net in Base 4 — Upper bound on s

There is no (128, 196, 13310)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10111 908407 061651 683750 811615 124071 188891 144888 994600 886628 024205 389475 490049 729977 801196 247901 552334 171688 415639 515310 > 4¹⁹⁶ [i]